17253054 Math Integration

8/8/2019 17253054 Math Integration

http://slidepdf.com/reader/full/17253054-math-integration 1/20

LEARNING OBJECTIVESLEARNING OBJECTIVES

To understand the virtual loss of GM andTo understand the virtual loss of GM andthe calculations.the calculations.

To calculate the maximum trim allowedTo calculate the maximum trim allowedto maintain a minimum stated GM.to maintain a minimum stated GM.

To understand the safe requirements forTo understand the safe requirements fora ship prior enter into dry dock.a ship prior enter into dry dock. ToTo

understand the critical period during dryunderstand the critical period during drydocking process.docking process.

Font: Verdana, bold

Size: Min. 24

Slide design & Background:

Clear & Contrast



Course Outline

Name of Course :Chief and Second

Engineer 3000 kW or more (Unlimited

Voyage) Course Code/Module : ECSU , Part

A

Subject : Mathematics and

Engineering Drawing



Course Outline

Module Aims

To provide students with the familiarization to the fundamentals of calculus Mathematicsrequired for engineering practice and problem solving.

General Learning Objective - GLO

Recognize that integration is the inverse process of differentiation, and apply this knowledgeto determine the area/volume/work done.

Specific Learning Objectives - SLO Recognize that integration can be considered the reverse of differentiation

process.

Explain the integration of x, trigo. functions, 1/x, exponential functions.

Evaluate the constant of integration.

Perform the definite Integral.

Apply integration to find:

a. Area under curves. Volume of solid revolution

Work done

Mean & root mean square (rms) values

Centroid



Course outline Instructional Hours

Lecture : 40 hours

Topics Hours

Integration as reverse of differentiation 2

Integration of functions: x,Trig,1/x, Exponential 8

Evaluation of constant of integration 4

Definite integral 6

Application of integral calculus to: 20

a. Area under curves.

b. Volume of solid revolutionc. Work done

d. Mean & root mean square (rms)

e. valuesCentroid20



Course Outline

Integration as the Process of Summation

Integration as the Reverse of Differentiation

Integration of functions Applications of Integration : Areas Bounded

by Curves and Volumes of Revolution



Teaching Methods -

Combination of combination of methods as necessary -lectures, practice

Assessment Methods

Lecturer Class Assessment 1 20 % Lecturer Class Assessment 2 20 %

Lecturer Class Assessment 3 20 %

Final Exam 40 %

RecommendedT

exts K A Stroud (1992), Engineering Mathematics ProgrammesAnd Problems

G.S.Sharma & I.J.S.Sarna (1992), Engineering Mathematics



We know how to find the area of simple

geometric shapes such as the triangle below

Integration : Concept and Theory

y

x

21

1

2



But how do we find the are of geometric object

which do not have straight edges ?

ba

x

y

f(x)




So, how do we go about finding the area under

the curve f(x), between x=a and x=b ?

Well, we can divide the area under the curve into

separate rectangles «

« find the area of each rectangle «

« and then sum these areas in order to find an

approximate answer to area under curve




Find area of each rectangle «

« then sum all areas between x=a and x=b


ba

x

y

f(x)

h



Process of Integration

Integration is reverse of differentiation

In differentiation, if f(x)= then f`(x)= 4x . Thus the integral of

integration is the process of moving from f (̀x) to f(x). By similar

reasoning, the integral of.

Integration is a process of summation or adding parts together and an

elongated S, shown as, is used to replace the words µthe integral of¶. Hence,

from above,

µc¶ is called the arbitrary constant of integration



Integrationis the reverse process of differentiation.

Power series integration,

increase the exponent by one

- (P NO2 )-n+1

and divide by the

new exponent.

-n+1

k t

1

= + C

Constant

of integration

- (P NO2 )-n dP NO2 = k dt



The Fundamental Theorem of Calculus

If then f x dx F b F aa

b ( ) = ( ) − ( )∫ .

F x f x' ,( ) = ( )

If we know an anti-derivative, we can use it to find the

value of the definite integral.





Integration of function

The general solution of integrals of the form

axnd x, where a and n are constants is given by:

This rule is true when n is f r actional, zero, or a

positive or negative integer , with the exception of n = -1.

).1( 1

1

{!

´ nC n

x

dx x

n

n





Standard Integrals



The integral of constant K is kx+c. for

example

Integr al of sever al terms = Sum of integr al of the separ ate terms

f or exemple



Problems



Exercise

Documents

17253054 Math Integration